On Nonexistence of Baras-Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations

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Abstract

The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B-1 subset of R-N, N >= 3, u(t) = Delta u + c/vertical bar x vertical bar(2)u in B-1 x (0, T), u vertical bar(partial derivative B1) = 0, in the supercritical range c > c(Hardy) = (N-2/2)(2) does not have a solution for any nontrivial L-1 initial data u(0)(x) >= 0 in B-1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u(n) (x, t)} of classical solutions corresponding to truncated bounded potentials given by V (x) = c/vertical bar x vertical bar(2) (bar right arrow) V-n(x) = min{c/vertical bar x vertical bar(2), N} (n >= 1) diverges; i.e., as n -> infinity, u(n)(x, t) -> +infinity in B-1 x (0, T). Similar features of "nonexistence via approximation" for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u(0)(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u(t) = Delta(vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(p-1) u/vertical bar x vertical bar(2), m >= 1, p > 1, and u(t) = -Delta(2)u + c/vertical bar x vertical bar(4)u , c > c(H) = [N(N-4)/4](2), N > 4.
Original languageEnglish
Pages (from-to)123-143
Number of pages21
JournalProceedings of the Steklov Institute of Mathematics
Volume260
Issue number1
DOIs
Publication statusPublished - 2008

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Nonlinear Parabolic Equations
Positivity
Nonexistence
Vertical
Approximation
Infinity
Nonlinear Parabolic Problems
Partial derivative
Diverge
Classical Solution
Unit ball
Semilinear
Heat Equation
Justify

Cite this

@article{9acfe4ad067d4a44bf8a3a3ee46db432,
title = "On Nonexistence of Baras-Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations",
abstract = "The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B-1 subset of R-N, N >= 3, u(t) = Delta u + c/vertical bar x vertical bar(2)u in B-1 x (0, T), u vertical bar(partial derivative B1) = 0, in the supercritical range c > c(Hardy) = (N-2/2)(2) does not have a solution for any nontrivial L-1 initial data u(0)(x) >= 0 in B-1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u(n) (x, t)} of classical solutions corresponding to truncated bounded potentials given by V (x) = c/vertical bar x vertical bar(2) (bar right arrow) V-n(x) = min{c/vertical bar x vertical bar(2), N} (n >= 1) diverges; i.e., as n -> infinity, u(n)(x, t) -> +infinity in B-1 x (0, T). Similar features of {"}nonexistence via approximation{"} for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u(0)(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u(t) = Delta(vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(p-1) u/vertical bar x vertical bar(2), m >= 1, p > 1, and u(t) = -Delta(2)u + c/vertical bar x vertical bar(4)u , c > c(H) = [N(N-4)/4](2), N > 4.",
author = "Galaktionov, {Victor A}",
year = "2008",
doi = "10.1134/S0081543808010094",
language = "English",
volume = "260",
pages = "123--143",
journal = "Proceedings of the Steklov Institute of Mathematics",
issn = "0081-5438",
publisher = "Maik Nauka-Interperiodica Publishing",
number = "1",

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TY - JOUR

T1 - On Nonexistence of Baras-Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations

AU - Galaktionov, Victor A

PY - 2008

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N2 - The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B-1 subset of R-N, N >= 3, u(t) = Delta u + c/vertical bar x vertical bar(2)u in B-1 x (0, T), u vertical bar(partial derivative B1) = 0, in the supercritical range c > c(Hardy) = (N-2/2)(2) does not have a solution for any nontrivial L-1 initial data u(0)(x) >= 0 in B-1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u(n) (x, t)} of classical solutions corresponding to truncated bounded potentials given by V (x) = c/vertical bar x vertical bar(2) (bar right arrow) V-n(x) = min{c/vertical bar x vertical bar(2), N} (n >= 1) diverges; i.e., as n -> infinity, u(n)(x, t) -> +infinity in B-1 x (0, T). Similar features of "nonexistence via approximation" for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u(0)(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u(t) = Delta(vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(p-1) u/vertical bar x vertical bar(2), m >= 1, p > 1, and u(t) = -Delta(2)u + c/vertical bar x vertical bar(4)u , c > c(H) = [N(N-4)/4](2), N > 4.

AB - The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B-1 subset of R-N, N >= 3, u(t) = Delta u + c/vertical bar x vertical bar(2)u in B-1 x (0, T), u vertical bar(partial derivative B1) = 0, in the supercritical range c > c(Hardy) = (N-2/2)(2) does not have a solution for any nontrivial L-1 initial data u(0)(x) >= 0 in B-1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u(n) (x, t)} of classical solutions corresponding to truncated bounded potentials given by V (x) = c/vertical bar x vertical bar(2) (bar right arrow) V-n(x) = min{c/vertical bar x vertical bar(2), N} (n >= 1) diverges; i.e., as n -> infinity, u(n)(x, t) -> +infinity in B-1 x (0, T). Similar features of "nonexistence via approximation" for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u(0)(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u(t) = Delta(vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(p-1) u/vertical bar x vertical bar(2), m >= 1, p > 1, and u(t) = -Delta(2)u + c/vertical bar x vertical bar(4)u , c > c(H) = [N(N-4)/4](2), N > 4.

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